High School Physics Intro to Waves

Up until now we’ve dealt with linear movement in mechanics and a bit of circular motion, however there is another kind of movement that we need to discuss before we jump into waves. Think of a spring and how, after released from compression, it enters a rhythm of rising up and falling down. The only thing that stops it from doing this continuously forever is friction. This kind of rhythmic motion is called a periodic motion, and more specifically, a simple harmonic motion. This term will come up again later in the course, so be sure to keep it in mind as we progress.

At 22:50, how you get 2pi/lambda(x) - 2pi/T - 2pi.Specifically I don't understand where you get the 2pi at the end from.Please explain..

2 answers

Last reply by: Anna HaSun May 31, 2015 8:29 PM

Post by Anna Haon May 31, 2015

Hi Professor Selhorst-Jones,

Thank you for your wonderful videos! They have been very helpful :)

I was just wondering do transverse and longitudinal waves require a medium? I also wanted to check that: mechanical waves require a medium and can be transverse and longitudinal waves. And electromagnetic waves do not require a medium and are only transverse waves. Right?

Thank you!

1 answer

Last reply by: Professor Selhorst-JonesMon Sep 9, 2013 11:28 AM

Post by HARRISON IGWEon September 9, 2013

hi professor I've been finding difficulties in solving this problem .The equation Y=55m(3x-4t),where Y is mm,X is in m,T is in seconds represent a wave motion . Determine (i)frequency (ii)period (iii)speed of wave?

2 answers

Last reply by: Goutam DasTue Jun 11, 2013 3:23 AM

Post by Goutam Dason June 7, 2013

Hi professor, as far as I know,All waves need a medium to propagate.

The medium of sound is a fluid, and fluids have no shear strength, therefor they can only be longitudinal (pressure) waves.

Electromagnetic waves are transverse (shear) waves, so their medium (ether) has to be solid. The formula for the speed of a transverse wave in a solid isc = âˆš(G/Ï), where G is the shear modulus (stiffness) and Ï is the inertial density of the medium. If we knew the density of the ether, we could calculate its shear modulus from the known speed of light, and vice versa. Evidently, the ether is ultra denseâ€”perhaps a googol times denser than a neutron star. That's inertial density; the ether probably has no gravity of its own, as (I believe) it is also the medium of gravity, and all the other forces. What do you think professor?

Intro to Waves

There are three main categories of waves:

Mechanical Waves: These travel through a material medium.

Electromagnetic Waves: EM waves do not require a material medium to exist.

Matter or Quantum Mechanical Waves: These describe the motion of elemental particles (electrons, protons, etc.) on the atomic level. We won't investigate them in this course.

We also classify waves based on how they move:

Transverse Waves: The particles of the wave move perpendicular to the motion of the wave.

Longitudinal Waves: The particles of the wave move parallel to the motion of the wave. This is done through compression and rarefaction (expansion), i.e., the wave is transmitted by pressure changes.

We describe a wave with the following characteristics:

Amplitude (A): How tall the wave is at its maximum height.

Wavelength (λ): The distance between "repeating" points on the wave, such as top-to-top.

Wave speed (v): How fast the wave is moving.

Period (T): The time it takes to go through a full oscillation.

Frequency (f): The number of oscillations that occur per second. [The unit for this is the hertz (Hz) where 1 Hz = [1/1s]. Thus, f = [1/T] and T = [1/f].]

Because speed, frequency, and wavelength are all related, v = λf .

We can find the height (or pressure differential if it's a longitudinal wave) with the following equation:

y(x,t) = Asin(kx − ωt).

x is the horizontal location we are considering.

t is the time we are looking at the wave.

k is the angular wave number and is connected to the wavelength:

k =

2π

λ

.

ω is the angular frequency and is connected to the period (which is connected to the frequency):

ω =

2π

T

⇔ ω = 2π·f.

Intro to Waves

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

Transcription: Intro to Waves

Welcome back to Educator.com Today we’re going to be talking about an introduction to waves.0000

We’re going to finally get an understanding of how waves work and they have an incredibly large presence in our daily lives.0005

Waves are way for one object or location to have energy transferred to another object or location.0011

That seems like a really large definition and it is. It’s not the only way for energy to be transferred but it a motion of energy from location to location.0018

It also has a lot of other effects and we’ll be investigating some of those.0028

There are many different kinds of waves and you’re constantly around them.0032

Some basic examples that you’re being currently being exposed to: sound and light.0036

Also any vibrating strings, waves in water, and the list goes way on.0041

There’s all sorts of other things, seismic waves in the ground, vibration along a steel pipe.0046

There’s quantum motion in waves, though we won’t be getting into that.0052

Waves make up a fundamental part of the universe and nature around us.0055

To begin with, let’s imagine the idea of a pulse. Imagine you’ve tied one end of a string to a wall, so it’s tied over here and you pull the string taunt.0061

Then you whip it, really suddenly, once. You whip it hard, what’s going to happen?0072

Well you’re going to create a pulse of energy that’s going to be sent down the string.0077

As the energy might be here, we’ll have a sort of raised up area and then as time goes on, not long time probably, it’s going to move to here and then it’ll move to here then it’ll move to here until eventually it hits the wall.0081

So that pulse of energy will be sent down the string and it’ll keep moving down the string towards the wall.0095

We whip it once and we move the energy, the energy that we put into our whip gets transmitted down the string.0099

Some of it goes into heat and other things, we won’t actually be investigating the energy, we will just be investigating the motion of waves.0105

There is definitely a change of energy from a location to another location.0111

Finally, matter or quantum mechanical waves. These are waves that describe the motion of elemental particles, like the electron, the proton, or even smaller, the quark or the photon on the atomic level.0284

They’re very important to modern physics. They have a whole big impact and what we understand now and where our research is going currently.0296

We’re not going to be investigating them in this course. They behave a little bit differently than the classical waves we’re going to be studying.0302

They’re going to be more challenging to understand, so that’s something we’re going to save a future course.0307

Alright, in addition to the previous categories, the mechanical waves, EM waves, and mater waves.0313

We won’t be talking about mater waves. Mechanical waves are the ones we most understand.0320

They’re the ones that have motion medium. EM waves, they sort of come with their own medium.0325

In addition to the previous categories, we can also classify waves based on how they move.0329

Transverse waves. The particles of the waves move perpendicular to the motion of the waves.0334

If we got something that would normally be a flat line like the surface of the sea.0339

Then we’ve got a wave, an undulating curve on top of it that causes the surface of that sea to move up and down depending on the location of the wave.0344

That’s going to be a transverse wave. It’s happening transverse to the motion, it’s happening perpendicular to the motion of the wave.0353

Examples include a vibrating string, EM waves, any waves in water; not any waves, sorry you can also have sound waves in water.0361

Waves that we’re used to seeing in water. There is a huge varieties of things like this.0369

If you were to knock on a pipe, you’d also once again…if you were to shake a pipe, you’d get waves. If you were to knock on a pipe, you’d get sound waves.0375

For velocities and meters per second, we take that and one second later it will be that much velocity meters ahead.0630

Remember the wave is moving along. And as this moves forward we’re going to wind up seeing more of it coming out of it.0638

There is always more wave that’s going to be put in. Either it’s too the left outside of where we’re looking or it gets whipped into place by the motion of whatever is creating the wave.0645

The wave continues out on either side, and the whole thing moves forward.0656

Period. The period is the time it takes to go through a full oscillation.0664

Remember if we got some speed V, that this whole thing has, we’re going to be able to get from here to here and we’ll have a repartition.0668

Now one way is we could change the location we’re looking, but we could also fix the location that we’re looking here.0678

We’d be able to see that T later. We start here, say we start here, but T later because of the velocity will wind up showing up there.0684

If you notice, the velocity times the period is always going to wind up equaling the wave length.0698

Because the amount of time that it takes, if the velocity goes this way and T later is how long it takes to go through a single oscillation, the period is how long it takes from wave peak to get to another wave peak.0708

Or one point on the wave to get to its corresponding twin point on the wave.0721

That’s going to wind up having to be the wave length right? That’s what the definition of the wave length is.0727

We can think of it as a movement, the time that we’ve allowed it to move, and has given us a repetition, a period.0732

Or we could think of it as how far different we’ve seen in the distance that we’re looking in the wave, has given us a repetition on the wave.0740

So we’ve got the velocity times the time that it takes for the period is going to equal that wave length because they’ve got to be equivalent.0747

That’s because the fact we’re looking for long it takes to get from one like point on the wave to another point.0866

How long it takes for a single oscillation to occur is going to also be if we divide one second by how long it takes we’ll get the number of times we get an oscillation.0873

If we have the number of times oscillation happens in a second and we divide one second by those numbers of oscillations, we’ll get how long it takes each oscillation to occur.0882

Finally because speed, frequency, and wave length are all deeply related, we’re going to have the fact that velocity is equal to lambda times frequency.0892

Which is equivalent to velocity times time equals lambda. Remember we know that velocity that we’re moving, times the time it takes to from peak to peak has to be equal to lambda.0900

We’ve got the same thing going on here since V=λf. Well if we divide by F on both sides, we get v/f=λ. Which 1/f=t, so we get vt=λ.0914

So these are equivalent expressions. Why is V=λf make sense? Well if we’ve got lambda distance here, this is lambda…distance.0926

And we know that time, in one second we’re going to see the frequency occur. So if we see ten oscillations in one second, then how much distance have we covered?0939

We’ve covered ten oscillations to each oscillation as one wave length, then ten times the wave length.0949

So the frequency, the number of oscillations we occur, that occur in one second, we have in one second, times how long each one of those oscillations is, is going to be the velocity since we can just divide by one second.0955

Velocity is equal to lambda, the distance, times frequency, which comes in one of our seconds. So we’ve got meters per second.0967

Great. Wave equation, finally we can talk about if want to know the height of a point on a wave.0975

We’ll need to know two inputs. The horizontal location x and the time we’re looking at the wave.0982

Here’s just a quick sketch of a wave. If we got some wave like this, we could either look at this location x or maybe we want to look at this location x.0987

We’re going to get totally different results for that. However, what happens if then we also say, one of them is T=1?0999

What happens when we look at T=2? Well then the entire wave is going to have slid over some amount.1006

We’re going to have the same wave, but it’s going to have slid over and we’re going to get totally different answers for each one of those x’s.1013

That means, the time that we’re looking and also the location on the wave that we’re looking matters.1021

We’ve got a two variable function, we’ve got an equation that’s going to be based off of two variables that give us that dependent variable, that why, what output value.1027

Notice that one other thing about this, if we only care about the point where the wave originates, the very beginning point, x=0, we can simplify this because we can just knock out that x term.1037

We can make it simple at A times sin of omega T. Now, at this point you’re probably wondering what k and omega is, that’s a great question, we’re about to answer it.1046

This part right here winds up being the exact same as this part right here.1371

The only part that’s different is this here, but remember since we’re dealing with sin, since we’ve moved just one more 2π, we’re some location.1379

Then we just spin it around and boom we wind up being at the exact same location because we’ve moved by -2π.1389

So this location here, this stuff all here, is our real location. So if wind up shifting things by one whole period, even if we start at some time that isn’t a whole thing of a period, the way we’ve got this set up.1396

Since we’re using sin, a regular periodic function, we’ve got the fact we’re able to make those oscillations occur in our mathematics, our algebra, is able to support our visual oscillations.1409

We move a period temporally in algebra, we wind up moving that same full length time, so we see the exact thing.1422

The exact same would happen if we used k, if we used…if we moved x around and this sort of thing, but this is just to illustrate that the math here is really working and why it is.1430

We’ve got to make sure that 2π over t is able to handle a motion of a period as causing no effect to the values we’ll get out. Same thing with the motions of a wave length.1439

That’s why that 2π comes in, is because sins, sins natural period is 2π right? Its period before it repeats is 2π.1448

So we have to have a way to have those two periods, the period of the wave we’re working with and the period of the natural function we’re working with to be able to communicate with each other and that’s what this is all about.1456

Then finally, if we have 2π/t since over t is the same thing as times frequency, we get omega is also equal to 2π times the frequency, as simple as that.1465

Finally we’re ready to work on some examples. If we have a CPU on some device that has a frequency of 1 gigahertz, then we’ve got a frequency equals 1 gigahertz.1476

How many hertz is that? Well it goes, kila, mega, giga. So 10^3, 10^6, 10^9.1487

Which becomes 10^-9, which is the same thing as mila, micro, nano. So we get one nanosecond is how long it takes.1523

There are 10^9 nanoseconds in a single second, which makes sense because 10^9, there has to be 10^9 nanoseconds because each of those cycles has to be complete for it to be able a whole frequency of 10^9 cycles completing.1524

So CPU, we’ve got the same, it’s not directly a wave in the same way, but we’ve got the fact that it’s having repetitive things happen.1539

Its going through cycles, it’s cycling through data. We’re able to talk about it in terms of frequency and in terms of periods just like we do with waves.1566

If the speed of light is velocity 3x10^8 meters per second, mighty fast. And you receive a wave with a wave length of 500 nanometers. Very small wave length. What is its frequency?1573

Speed of light is v=3x10^8 meters per second and you receive a wave with a wave length of 500 nanometers. We already know what v is, v is here.1588

So 500 nanometers is the wave length, right? The wave length equals 500 nanometers, which is the same thing as nana, 10^-9, 500x10^-9 meters.1598

If we want to know what its frequency is, we remember that velocity is equal to the frequency times lambda.1611

The frequency times lambda has to give us the velocity because that’s how much distance has been covered.1620

Wave length is a chuck of distance and frequency is how many times you have those chunks distance in one second to we cover. Frequency times wave length.1625

v=fxλ, so v/λ=f, so if we want to know what f is. F is going to be equal to 3x10^8/500x10^-9.1637

We plug that into a calculator and we get 6x10^14 hertz. That’s a giant frequency compared to the stuff that we’re used to seeing in sound.1655

For light though that’s pretty reasonable and you’re probably seeing a color pretty close to that.1666

6x10^14 hertz is a receivable for the human eye, it’s something in the visual spectrum and I think, don’t hold me to this, it’s probably pretty close to either yellow, green, or blue, somewhere in that sort of range.1671

Probably a little bit closer to the blue, green or blue. Anyway, that’s something that your eye is actually really able to see and so as opposed to seeing a numerical frequency data when we look at something.1687

We don’t say “Oh, that frequency is that”. We see a color. We’ve got these other ways of interpreting the information that the universe is sending to us.1697

It is a real thing, we are getting real information here just like when we go to some height, we’re really at a height but we can measure that height and periodically we’re able to measure the frequency we see in periodically.1704

If you’re driving a car going 30 meters per second and the car beings to run over a rumble strip, rumble strips are these evenly spaced grooves used to alert drivers, so little down grooves in the road like this.1716

So when a tire rolls over it, the tire falls in the up and down motion, winds up jostling the driver and they notice something, or they’re accidentally swerving off to the side, they notice that they’re swerving off to the side or if they’re coming up on some toll, they notice that they’re coming up to some toll.1733

It’s just something to alert drivers. If the strip vibrates your car at 98 hertz, what’s the spacing of the grooves?1747

First off, I’d like to point out that this isn’t technically a wave. Just like in #1 we weren’t technically working with a wave, but we can still apply many of the concepts.1754

In this case, the wave speed, we know the velocity of the car is equal to 30 meters per second.1762

While the wave might not be moving, the way it’s experiencing the wave is moving. In another way we could consider the tire is still, and the wave is the thing moving.1771

From your point of view you can’t really tell who’s moving when you’re inside of the car, although we look at our scenery and we can easily tell.1783

But you know, we can’t be sure who’s the thing that’s moving. It’s possible the road is the thing moving. So you can take it as the wave moving underneath you.1789

Once again, we’re not getting full repartition quite the same, maybe. But we’re not necessarily thinking it purely in terms of wave lengths, but that isn’t the issue.1796

So we’re running over some rumble strips and it’s moving by us at 30 meters per second, either because the cars moving, or because it’s moving relative to us.1810

If the strip vibrates your car or bounces you up and down 98 hertz or 98 times per second, what the spacing of the grooves got to be?1818

It’s the exact same thing, if it manages to bump us up and down 98 times in a second and the wave length is the spacing of the strips. The spacing of the grooves is going to be the wave length.1827

How far they spread apart, then the 30 meters per second that we experience has got to be that number of times that the wave length times how many times they show up in a second.1838

Velocity is equal to the frequency times lambda. So we plug in our numbers and if we want to know what lambda, is we’re going to have the velocity divided by the frequency.1849

We plug in our numbers 30 meters per second divided by a frequency of 98 hertz and we get 0.306 meters.1860

There we are, the spacing of those grooves is about a third of a meter which makes a lot of sense if you’ve ever looked at them on the street.1872

Example four. Use the following diagram to give an equation describing the wave in the diagram.1882

Now to begin with, lets point out, we just want to remember y equals a, the amplitude, times sin, the function that allows us to have periodicity, allows us to have oscillations occur algebraically.1887

kx - ω x t. So x, the location we are in the wave. T the time that we’re looking at the wave. Omega is the factor that allows us to handle the fact that periods cause repetitions.1901

K is the fact that allows us to handle that wave lengths cause repetitions. To begin with, we know that the period is equal to 0.02 seconds.1915

We know that amplitude is equal 0.5 meters. Okay, great. What is this here?1927

Well that is not equal to the wave length. Remember, if we look just to the right of this point, and look just to the right of this point. We should see the exact same thing.1935

We don’t see the exact same thing, it’s going down over. If we go over here though, we will see it.1944

And because its sin wave, it’s evenly spaced out throughout, we know that 3.5 meters isn’t going to be the wave length, but it’s going to be half the wave length because it’s in one of the dips.1950

So the wave length over 2, so that means that our wave length is equal to 7 meters.1961

That’s all the data that we wind up needing. Now we want to solve for what k has to be.1968

K is equal to 2π/λ, so k is equal to 2π/7. Omega is equal 2π over the period.1972

By the way, k, we’re throwing around k a lot. I never mentioned this but k is not the same k as when we’re dealing with springs. It’s a different k, we’re using it to mean a totally different thing at this point.1985

The k that we’re using in this stuff is different than the k we used before. Just like how little t and big t don’t necessarily mean the same thing. We can even wind up using the same letter for different things, and we have know contextually what we’re talking about.1999

Sorry, I made that assumption, but that’s actually a really important thing. You don’t want to get confused and think that springs and waves necessarily have to do with each other every time. It’s not that same spring constant we were talking about before.2011

Totally different use of k. Anyway back to the problem. Omega is equal to 2π divided by the period.2024

If we know the period is, 2π/0.02 seconds and that will wind up giving us 100π.2030

Simple as this at this point, we just plug in all of our numbers. Y is equal to that amplitude, 0.5 meters times sin, of the numbers 2π/7x-ω100π times the time that we’re looking.2039

That right here is our answer. Simple as that. So we just want to be able to analyze the diagram.2065

One of the most important things to pay attention to the fact that wave length has to be not just some distance where you get the same point, but some distance where you get a point that means the exact same thing.2071

That if you look just a little bit further on and a little bit further behind you’re going to see a full repetition.2083

It’s not just the same point, because the same point occurs at any horizontal thing, except for the very tops and bottoms.2089

These pairs of points are not enough to determine a wave length. What you need is to reach is to reach a little bit farther and look here.2097

The very top to top because tops themselves only occur once every wave length.2104

Whereas middles occur twice. Same with bottoms, you can go from the bottom to the bottom and the top to the top.2111

That’s normally the easiest thing to measure, but if you’re measuring from the middle to the middle, this isn’t enough.2117

You need to also go to here. Okay, great. I hope you enjoyed that, I hope that made sense, waves are a whole bunch of big ideas, but we’re going to…2123

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